| L(s) = 1 | − 158·2-s + 4.37e3·3-s + 3.95e3·4-s − 1.56e5·5-s − 6.91e5·6-s − 3.95e5·7-s − 2.48e6·8-s + 1.43e7·9-s + 2.46e7·10-s − 1.02e8·11-s + 1.73e7·12-s + 3.25e8·13-s + 6.24e7·14-s − 6.83e8·15-s + 5.35e7·16-s − 2.96e9·17-s − 2.26e9·18-s + 5.10e9·19-s − 6.18e8·20-s − 1.72e9·21-s + 1.61e10·22-s − 2.91e10·23-s − 1.08e10·24-s + 1.83e10·25-s − 5.13e10·26-s + 4.18e10·27-s − 1.56e9·28-s + ⋯ |
| L(s) = 1 | − 0.872·2-s + 1.15·3-s + 0.120·4-s − 0.894·5-s − 1.00·6-s − 0.181·7-s − 0.418·8-s + 9-s + 0.780·10-s − 1.58·11-s + 0.139·12-s + 1.43·13-s + 0.158·14-s − 1.03·15-s + 0.0498·16-s − 1.75·17-s − 0.872·18-s + 1.31·19-s − 0.107·20-s − 0.209·21-s + 1.38·22-s − 1.78·23-s − 0.483·24-s + 3/5·25-s − 1.25·26-s + 0.769·27-s − 0.0218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{7} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 79 p T + 1313 p^{4} T^{2} + 79 p^{16} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1152 p^{3} T + 150475443134 p^{2} T^{2} + 1152 p^{18} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 102396512 T + 631846058204434 p T^{2} + 102396512 p^{15} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 325165772 T + 7717217329715862 p T^{2} - 325165772 p^{15} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2967842012 T + 7829866235445539222 T^{2} + 2967842012 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5106599936 T + 27227327218576253478 T^{2} - 5106599936 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 29151569592 T + \)\(55\!\cdots\!54\)\( T^{2} + 29151569592 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 107462175868 T + \)\(16\!\cdots\!38\)\( T^{2} + 107462175868 p^{15} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 198740052648 T + \)\(53\!\cdots\!02\)\( T^{2} + 198740052648 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1056630568916 T + \)\(92\!\cdots\!06\)\( T^{2} + 1056630568916 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1396159784668 T + \)\(14\!\cdots\!62\)\( T^{2} + 1396159784668 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2151248940200 T + \)\(37\!\cdots\!50\)\( T^{2} - 2151248940200 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4140187105720 T + \)\(17\!\cdots\!90\)\( T^{2} - 4140187105720 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 874168845748 T + \)\(72\!\cdots\!14\)\( T^{2} - 874168845748 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 33007548610976 T + \)\(94\!\cdots\!58\)\( T^{2} + 33007548610976 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 46937748465260 T + \)\(17\!\cdots\!18\)\( T^{2} - 46937748465260 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 101655323829912 T + \)\(62\!\cdots\!22\)\( T^{2} + 101655323829912 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 69353886599296 T + \)\(70\!\cdots\!06\)\( T^{2} + 69353886599296 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4916861497748 T - \)\(18\!\cdots\!26\)\( T^{2} - 4916861497748 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 341880895398600 T + \)\(71\!\cdots\!98\)\( T^{2} - 341880895398600 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 248949501567096 T + \)\(11\!\cdots\!62\)\( T^{2} + 248949501567096 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 274264939790196 T + \)\(27\!\cdots\!58\)\( T^{2} - 274264939790196 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 779932726116868 T + \)\(12\!\cdots\!42\)\( T^{2} - 779932726116868 p^{15} T^{3} + p^{30} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.69271526918795188241987355743, −14.83673711802478605856237803272, −13.62947151585796485229643817075, −13.56277597533997016723489582408, −12.62570859959802615760865918893, −11.69365629183776210696189333261, −10.86805197732118773570226995869, −10.23884881034412563953526882882, −9.058046507959379208478480967739, −8.961540834624156573986847304293, −8.000569979755315446608278773973, −7.64290983678448896173115535161, −6.58791342265262177251975704369, −5.36856069582192938032725212403, −4.08215058863593950631948996602, −3.43372242841716925283490436972, −2.50047693339992591167108000907, −1.56501555782333312247740380858, 0, 0,
1.56501555782333312247740380858, 2.50047693339992591167108000907, 3.43372242841716925283490436972, 4.08215058863593950631948996602, 5.36856069582192938032725212403, 6.58791342265262177251975704369, 7.64290983678448896173115535161, 8.000569979755315446608278773973, 8.961540834624156573986847304293, 9.058046507959379208478480967739, 10.23884881034412563953526882882, 10.86805197732118773570226995869, 11.69365629183776210696189333261, 12.62570859959802615760865918893, 13.56277597533997016723489582408, 13.62947151585796485229643817075, 14.83673711802478605856237803272, 15.69271526918795188241987355743
